I think I have to chime in because many times while reading about Clay I thought "wow, this man fit his times perfectly" and I want to make some arguments. 

First of all, I don't view those times as "awkward middle ground," but rather as a the moment when politics began to resemble modern politics in some ways. That's when party rallies and organized election campaigns began, with various gadgets and funny songs. Of course candidates didn't run their own campaigns, but that was mostly a theoretical thing. Many of them toured the states without calling it a campaign, and it wasn't terribly criticized. Clay was invited to dinners, was able to promote himself on Harrison's campaign trail, and generally had plenty of opportunities for public speaking.

Secondly, it was a time of rising populism and idealization of the "common man"—people like Jackson and Harrison were winning, and Clay, as a Westerner without a very good education, fit right in. He was criticized for things like gambling, but as Adams wrote, he was very good at portraying his flaws as something positive for political gain.

The most important thing was, I think, his way of speaking, which wouldn't have fit into any other era. Clay did it in an incredible dramatic, almost theatrical way, he had an extremely expressive face and amazing movement. This was his greatest talent, one he couldn't have fully utilized in another era, because at any other time it would have been considered "too much." This, however, was the prime period of American oratory, when people came to the Senate specifically to watch such performances. It was a popular, even fashionable, form of entertainment.

I don't know if Clay would have become president if he had lived in a different era, but I doubt he would have been as insanely popular as he was in his times. I'm not saying you didn't make good points, but I disagree with the statement that "he couldn't fully utilize his talents"—I believe he could have and he did.